INVARIANT TWINS
it must have been ripe port, to judge by what Sylvester got out
of the decanter.
Cayley and Sylvester came together again professionally
when Cayley accepted an invitation to lecture at Johns Hopkins
for half a year in 1881-82. He chose Abelian functions, in which
he was researching at the time, as his topic, and the sixty-seven-
year-old Sylvester faithfully attended every lecture of Ms
famous friend. Sylvester had still several prolific years ahead of
Mm, Cayley not quite so many.
We shall now briefly describe three of Cayley's outstanding
contributions to mathematics in addition to his work on the
theory of algebraic invariants. It has already been mentioned
that he invented the theory of matrices, the geometry of space
of n dimensions, and that one of his ideas in geometry threw a
new light (hi Klein's hands) on non-Euclidean geometry. We
shall begin with the last because it is the hardest.
Desargues, Pascal, Poncelet, and others had created projec-
tive geometry (see chapters 5,13) in which the object is to dis-
cover those properties of figures which are invariant under
projection. Measurements - sizes of angles, lengths of lines -
and theorems which depend upon measurement, as for example
the Pythagorean proposition that the square on the longest side
of a right angle is equal to the sum of the squares on the other
two sides, are not projective but metrical, and are not handled
by ordinary projective geometry. It was one of Cayley's greatest
achievements in geometry to transcend the barrier which,
before he leapt it, had separated projective from metrical pro-
perties of figures. From his higher point of view metrical geo-
metry also became projective, and the great power and flexi-
bility of projective methods were shown to be applicable, by
the introduction of 'imaginary' elements (for instance points
whose co-ordinates involve V — 1) to metrical properties.
Anyone who has done any analytic geometry will recall that
two circles intersect in four points, two of which are always
*imaginary'. (There are cases of apparent exception, for
example concentric circles, but this is close enough for our
purpose.) The fundamental notions in metrical geometry are the
distance between two points and the angle between two lines.
M.H.—VOL. n. K 433 *